Near-Horizon Conformal Symmetry and Black Hole Entropy in Any Dimension

TL;DR

This paper critiques and refines Carlip's near-horizon conformal symmetry approach to black hole entropy, extending it to higher dimensions and highlighting the importance of basis choice for accurate entropy calculation.

Contribution

It corrects the algebraic issues in Carlip's method and generalizes the derivation of black hole entropy to any dimension within Einstein gravity.

Findings

The algebra of conformal transformations is ill-defined on the horizon in Carlip's original approach.

Choosing a regular basis of conformal transformations yields a non-zero entropy proportional to horizon area.

The generalized derivation applies to higher-dimensional black holes, linking entropy to horizon area and surface gravity.

Abstract

Recently, Carlip proposed a derivation of the entropy of the two-dimensional dilatonic black hole by investigating the Virasoro algebra associated with a newly introduced near-horizon conformal symmetry. We point out not only that the algebra of these conformal transformations is not well defined on the horizon, but also that the correct use of the eigenvalue of the operator $L_0$ yields vanishing entropy. It has been shown that these problems can be resolved by choosing a different basis of the conformal transformations which is regular even at the horizon. We also show the generalization of Carlip's derivation to any higher dimensional case in pure Einstein gravity. The entropy obtained is proportional to the area of the event horizon, but it also depends linearly on the product of the surface gravity and the parameter length of a horizon segment in consideration. We finally point out that this derivation of black hole entropy is quite different from the ones proposed so far, and several features of this method and some open issues are also discussed.